A new variable step-size hybrid block method for the numerical solution of ordinary differential equations
*Ahmad Umar AbubakarCorresponding authorvtd1478@veltech.edu.inDepartment of MathematicsAvadiVel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and TechnologyChennai, Tamil Nadu, 600062, IndiaView full profile → , E. Chandrasekarandrchandrasekarane@veltech.edu.inDepartment of MathematicsAvadiVel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and TechnologyChennai, Tamil Nadu, 600062, India0000−0001−9393−4184View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Nov 2025
- Published Online:
- 09 Apr 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2412
- Pages:
- 987–1000
Abstract
Keywords
Subject Classifications
References
[1] N. Abasi, M. B. Suleiman, F. Ismail, Z. B. Ibrahim, H. Musa, and N. Abbasi, “A new formula of variable step 3-point block BDF method for solving stiff ODEs,” Journal of Pure and Applied Mathematics: Advances and Applications, vol. 12, no. 1, pp. 49–76 (2014).
[2] C. F. Curtiss and J. O. Hirschfelder, “Integration of stiff equations,” Proceedings of the National Academy of Sciences, vol. 38, no. 3, pp. 235–243 (1952).
[3] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, NJ, USA: Prentice-Hall (1971).
[4] Z. B. Ibrahim, K. I. Othman, and M. Suleiman, “Implicit r-point block backward differentiation formula for solving first-order stiff ODEs,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 558–565 (Mar. 2007), doi: 10.1016/j.amc.2006.07.116.
[5] Z. B. Ibrahim, K. I. Othman, and M. Suleiman, “Variable step block backward differentiation formula for solving first-order stiff ODEs,” in Proc. World Congress on Engineering (WCE), vol. II, London, U.K., Jul. 2–4 (2007).
[6] J. D. Lambert, Computational Methods in Ordinary Differential Equations. New York, NY, USA: Wiley (1973).
[7] M. B. Suleiman, H. Musa, F. Ismail, N. Senu, and Z. B. Ibrahim, “A new superclass of block backward differentiation formula for stiff ordinary differential equations,” Asian-European Journal of Mathematics, vol. 7, no. 1 (2014), doi: 10.1142/S1793557113500344.
[8] I. S. Mohd Zawawi, Z. B. Ibrahim, and K. I. Othman, “Derivation of diagonally implicit block backward differentiation formulas for solving stiff initial value problems,” Mathematical Problems in Engineering, vol. 2015, no. 1, Art. no. 179231 (2015), doi: 10.1155/2015/179231.
[9] R. K. P. Reshma, S. Prasad, C. K. Sharma, A. Deshpande, S. B. Bagal, and K. Tejaswi, “High-accuracy Runge-Kutta methods for stiff ODE systems,” Journal of Interdisciplinary Mathematics, vol. 28, no. 6, pp. 2217–2225 (2025), doi: 10.47974/JIM-2363.
[10] Q. A. Dang, T. H. Nguyen, and V. Q. Vu, “Eighth order numerical method for solving second order nonlinear BVPs and applications,” Journal of Applied Mathematics and Computation, vol. 71, no. 3, pp. 3577–3600 (2025), doi: 10.1007/s12190-025-02368-5.
[11] B. G. Ogunware, J. O. Kuboye, O. E. Abolarin, and C. O. Mmaduakor, “Half-step numerical model of order ten for solution of first and second orders ordinary differential equations,” Journal of Interdisciplinary Mathematics, vol. 25, no. 4, pp. 1023–1036 (2022), doi: 10.1080/09720502.2021.1887620.
[12] J. O. Kuboye and E. O. Adeyefa, “New developed numerical formula for solution of first and higher order ordinary differential equations,” Journal of Interdisciplinary Mathematics, vol. 25, no. 2, pp. 395–414 (2022), doi: 10.1080/09720502.2021.1925453.
[13] V. Murugesh, M. Priyadharshini, Y. K. Sharma, U. K. Lilhore, R. Alroobaea, H. Alsufyani, A. M. Baqasah, and S. Simaiya, “A novel hybrid framework for efficient higher order ODE solvers using neural networks and block methods,” Scientific Reports, vol. 15, Art. no. 8456 (2025), doi: 10.1038/s41598-025-8456-0.
[14] K. A. Hussain, “A new embedded improved Runge-Kutta approach to solve first order ordinary differential equation,” Journal of Interdisciplinary Mathematics, vol. 28, no. 3-B, pp. 1247–1254 (2025), doi: 10.47974/JIM-2214.




